Introduction to Probability

Hello. My name is Neil Cheetah. I'm an associate professor at Durham University.

In this course, we're going to be talking about probability.

So what is probability?

So probability is the mathematical discipline studying randomness or chance?

It has wide ranging applications in many areas,

even where you do not think there's any randomness present.

For instance, machine learning, biology, statistics and physics

at, to name a few.

So here is a motivating example from genetics.

Suppose we have two parents that are heterosexual capital, a little a individuals.

The chance that they each give their offspring a little A is 50%.

This means we can work out the chance that the offspring is little, a little a,

which is 25% chance.

This simple example illustrates why probability is useful.

In fact, it becomes really useful when the systems size becomes very large.

Probability is about estimating or calculating how likely

or probable an event is likely to happen.

An example of this. What is the chance that it will rain tomorrow?

In probability, we use a sliding scale from zero up to one

zero Are impossible events.

For example, picking a red ball from a bag containing only blue balls.

This is impossible,

and one is certain offence.

For example, picking a red ball from a bag containing only red bulls

along this sliding scale.

There is the value of half, and this represents an even chance.

For example, flipping a heads on a fair coin.

So some examples of probability,

the probability of observing heads after flipping a fair coin

equals one half

the probability of rolling

a fair die,

and it landing on six is equal to 1/6.

So why is this?

So the probability of an outcome or an event is

equal to the number of ways an outcome can occur,

divided by the number of possible different outcomes.

So if we go back to example one of the

probability of observing heads after flipping a fair coin,

there is one way for it to to land on their heads.

And there are two different possible outcomes, either heads or tails. So we get 1/2

to write this. What we do is we write p

of event, so here P stands for probability.

An event stands for the event that we're observing.

For instance, the event of observing heads after flipping a fair coin.

Let's look at a few examples,

so the probability of rolling one on a fair die is 1/6.

This is because there is only one possible way for the dice to land on a one,

and there are six different possible outcomes of the die.

Another example is the probability of rolling an even number on a fair die.

This is because there are three different possible outcomes.

Either it lands on 24 or six,

and the dye has six different possibilities.

So we get 3/6, which is equal to one half.

A third example is

the property of the dye landing on a prime number.

There are three different prime numbers that we can land on 23 and five.

And there are six different possible outcomes of where the dye can land.

So we get through over six.

1/4 example is the probability of getting dealt an

ace from a perfectly shuffled deck of cards.

There are four different aces in the pack, so that's four,

and there are 52 cards in a deck of cards, so we get 4/52.

So as a non examine double side remark. Probability is completely theoretical.

In fact,

there is a philosophical debate on how one should think about probabilities.

There are those are called frequentist, so

that think that probabilities made by repeating experiments multiple

times and the probabilities are given by relative frequencies.

Then there are those that are called subjective zor patients,

and they believe that probabilities should be interpreted as a guest made

before the event of how likely the event is to happen.

So an example of this would be weather forecasters.

So who's right? Who knows?

In probability, we just ignore such arguments.

This concludes

the mini Lecture on Introduction to Property.

In the next lecture,

we will look at frequency tables and frequency treats.